Extracavity pumped parametric Raman nanosecond crystalline anti-Stokes laser at 954 nm with collinear orthogonally polarized beam interaction at tangential phase matching

S. N. Smetanin; M. Jelínek; D. P. Tereshchenko; V. Kubeček

doi:10.1364/OE.26.022637

1. Introduction

Stimulated Raman scattering (SRS) in crystalline materials presents a simple and efficient method of nonlinear optical conversion of the laser frequency into the spectral domains where direct lasing is not possible or complicated. Such laser sources are called Raman lasers [1]. Moreover, using the parametric four-wave mixing of the Raman laser frequency components on the same resonant cubic nonlinearity of Raman scattering allowed generation of a collimated beam of not only Stokes, but also anti-Stokes components of the Raman radiation. Such lasers are called parametric Raman lasers [2–8]. In comparison with the conventional Raman (Stokes) lasers, the parametric Raman anti-Stokes lasers require phase matching fulfillment for the Stokes ↔ anti-Stokes parametric coupling. In the most cases, such phase matching in the Raman-active crystals was fulfilled by the non-collinear interaction of the laser pump and Raman radiation components [2–7], but conversion efficiency into the anti-Stokes component was low. For example, conversion efficiencies of 0.5% and 1.7% in the extracavity pumped parametric Raman KGd(WO₄)₂ and BaWO₄ anti-Stokes lasers were reported [5,6]. Theoretical analysis has shown that low conversion efficiency into the anti-Stokes component at non-collinear parametric Raman interaction was caused mainly by narrow angular tolerance of phase matching in comparison with the angular divergence of the interacting beams [2,3,8]. It is well-known from nonlinear optics of media with quadratic nonlinearity that using the birefringent nonlinear crystals can help to realize phase matching of three-wave mixing for collinear interaction of orthogonally polarized radiation components with controllable angular tolerance. Similar possibilities for four-wave mixing in cubically nonlinear Raman-active crystals were investigated for the first time in [9] at single pass SRS and later in [10] at intracavity SRS using the highly birefringent CaCO₃ crystal. Recently in [8] for the first time in nonlinear optics of media with cubic nonlinearity there was proposed to use tangential phase matching for four-wave mixing of the orthogonally polarized Raman laser radiation components in a birefringent Raman crystal with the widest angular tolerance at walk-off compensation of the interacting radiation components. In [8] the tangential phase matching was realized at single pass SRS in the CaCO₃ crystal under excitation by single laser source radiation split into the orthogonally polarized pump and probe beams with equal frequencies in contrast to the pioneer work [9] using two-color excitation. It allowed an increase in the optical-to-optical (532 nm to 503 nm) efficiency of the anti-Stokes generation up to 3.5% in the single-pass picosecond parametric Raman anti-Stokes convertor based on one CaCO₃ crystal with 1.4-μJ picosecond pulse output energy.

In this paper, we present new results of theoretical modeling and experimental study of anti-Stokes generation at 954 nm in the extracavity pumped parametric Raman CaCO₃ anti-Stokes laser with 1064-nm 6-ns laser excitation. Realization of tangential phase matching of collinear orthogonally polarized beam interaction in this laser allowed to achieve the highest optical-to-optical (1064 nm to 954 nm) efficiency of 4% at 0.3-mJ anti-Stokes output pulse energy among the known crystalline parametric Raman anti-Stokes lasers. In contrast to our previous work [8] we used here the extracavity laser configuration allowing the optimization of Raman components generation at nanosecond excitation. The increased excitation wavelength (1064 nm) gives also wider angular tolerance of tangential phase matching resulting in the anti-Stokes generation efficiency increase.

2. Tangential phase matching fulfillment

Tangential phase matching of four-wave-mixing anti-Stokes generation at SRS in a birefringent crystal was proposed by us recently in [8]. It represents an analogy of the known tangential phase matching of three-wave mixing in nonlinear optics of media with quadratic nonlinearity [11] that is demonstrated in Fig. 1.

Fig. 1 Tangential phase matching diagram (a) for difference frequency generation at three-wave mixing (idler, pump, and signal) of oeo-type (k_i^o = k_p^e – k_s^o) [11] and (b) for parametric Raman anti-Stokes generation at four-wave mixing (anti-Stokes, probe, pump, and Stokes) of eeoo-type (k_aS^e = k_p^e + k_p^o – k_S^o) [8] where Θ_pm is the phase-matching angle between the pump wave vector and the crystal optical axis, Δβ_opt is an optimal angle between the input waves.

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Tangentially phase-matched wave vector interaction is non-collinear with an optimum angle Δβ_opt between the input waves to obtain the locus of one input wave vector (the pump wave in Fig. 1) tangential to the locus of the generated wave vector (the idler wave in Fig. 1(a) or the anti-Stokes wave in Fig. 1(b)). This results in the widest angular tolerance of phase matching due to walk-off compensation for the pair of these waves (Poynting vectors P of these waves are coaxial). Note that the parametric Raman anti-Stokes frequency generation ω_aS = 2ω_p – ω_S (Fig. 1(b)) is similar to quadratically nonlinear difference frequency generation ω_i = ω_p – ω_s (Fig. 1(a)), but the angular tolerance of tangential phase matching shown in the diagram in Fig. 1(b) is wider. The locus curvatures for the $k_{p}^{o}$ and $k_{a S}^{e}$ wave vectors are close to each other in contrast to the diagram in Fig. 1(a) where the locus curvature for the $k_{p}^{e}$ wave vector is substantially lower than for the $k_{i}^{o}$ wave vector because $k_{i}^{o}$ is substantially shorter. It also explains why the pump and probe optical frequencies at the parametric Raman anti-Stokes generation should be equal (as in Fig. 1(b)) – to have the widest angular tolerance of tangential phase matching. This setup is in contrast to the case involving different frequencies of the pump and probe waves experimentally realized in the works [9,12–14] where the locus curvature for the anti-Stokes wave vector was substantially lower than for the pump wave vector. The anti-Stokes wave scattered parametrically from the probe wave in contrast to the Stokes wave generating by SRS from the pump wave. The lower locus curvature for the parametrically generated wave vector results in relatively low angular tolerance of tangential phase matching as shown also in Fig. 1. This explanation is also confirmed by the experimental results of active spectroscopy of Raman scattering of various media [13,14] where coherent scattering length was higher at lower difference between the pump and probe frequencies.

It is also necessary to note that the phase-matching type for efficient parametric Raman four-wave-mixing interaction for a negative uniaxial crystal should be eeoo-type (as in Fig. 1(b)) at λ_p < λ_d or ooee-type at λ_p > λ_d where λ_p is a pump wavelength and λ_d is a zero-dispersion wavelength of the crystal [10]. It is explained by that the most efficient parametric interaction of a pair of the equally polarized probe and anti-Stokes waves at the wavelengths of λ_p and $λ_{a S} = {(λ_{p}^{- 1} + ν_{R})}^{- 1}$ takes place when the Raman line with a vibrational frequency of ν_R of the crystal is coherently driven by the SRS interaction of a pair of the equally polarized (but orthogonally to the probe/anti-Stokes wave polarization) pump and Stokes waves at the wavelengths of λ_p and $λ_{S} = {(λ_{p}^{- 1} - ν_{R})}^{- 1}$ [8,10] resulting in the highest parametric coupling coefficient close to the Raman coupling coefficient [8,14].

In [8] we developed the procedure for determination of the eeoo-type phase-matching characteristics of the parametric Raman interaction (phase-matching angle, phase-matching angular tolerance, and walk-off compensation angles) in dependence on the angle Δβ between the pump and probe waves for negative uniaxial crystals and we applied it for a CaCO₃ crystal at λ_p = 532 nm. Now we apply it for a CaCO₃ crystal at λ_p = 1064 nm (< λ_d) using the refined data [15] of the Sellmeier equations for the IR region than the data used for visible region in [8,10]. The calculation results are presented in Fig. 2.

Fig. 2 Calculated dependences of (a) the phase-matching angle Θ_pm, the angular tolerance of phase matching ΔΘ_pm for the crystal length of L = 3.2 cm, and (b) the angles of walk-off compensation δβ_p_, _aS for the probe (δβ_p = Δβ – β_p) and anti-Stokes (δβ_aS = Θ_pm – Θ_aS – β_aS) waves on the angle Δβ between the pump and probe waves for the CaCO₃ crystal under pumping at the wavelength of λ_p = 1064 nm (β_p_, _aS are the walk-off angles for the probe and anti-Stokes extraordinary waves, Θ_aS is the angle between the anti-Stokes wave vector and the crystal optical axis).

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It can be seen that the collinear phase matching taking place at Δβ = 0 and Θ_pm = 6.08° is characterized by relatively low angular tolerance of ΔΘ_pm = 1 mrad and high uncompensated walk-off angle [8] for both the probe and anti-Stokes extraordinary waves of β_p_, _aS ≈24.6 mrad. On the contrary the tangential phase matching with the highest angular tolerance of ΔΘ_pm = 7 mrad takes place at Δβ_opt = 1.46° and Θ_pm = 6.9° that corresponds to full walk-off compensation for the anti-Stokes extraordinary wave (δβ_aS = 0), but walk-off compensation for the probe extraordinary wave is partial (δβ_p = 3.46 mrad). It slightly limits the effective interaction length by the value of L_δβ ≈d₀/δβ_p, where d₀ is the beam diameter. For example, at d₀ = 440 μm we get L_δβ = 12.7 cm, i.e. at the tangential phase matching we can use a long CaCO₃ crystal even for the strongly focused beam.

3. Laser system schematic

We propose a new scheme of the extracavity parametric Raman crystalline anti-Stokes laser presented in Fig. 3.

Fig. 3 Extracavity parametric Raman CaCO₃ anti-Stokes laser schematic.

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In contrast to the known extracavity parametric Raman anti-Stokes lasers [3,5,6] our laser was excited by two orthogonally polarized (pump and probe) laser beams with equal frequencies similarly as in the picosecond single-pass parametric Raman crystalline anti-Stokes converter reported recently by us in [8].

A c-cut CaCO₃ crystal with a length of L = 3.2 cm was used as the parametric Raman laser active crystal. This crystal was without any AR coatings because it would have an essential influence only on the extracavity Stokes SRS generation, but the anti-Stokes generation was single-pass. The incident pump wave polarization was oriented perpendicularly to the CaCO₃ crystal optical axis for ordinary wave pumping. At the pump wavelength of λ_p = 1064 nm, the anti-Stokes, 1^st Stokes, and 2^nd Stokes SRS components generating in the CaCO₃ crystal (ν_R = 1086 cm ^{– 1}) had wavelengths of $λ_{a S} = {(λ_{p}^{- 1} + ν_{R})}^{- 1}$ = 954 nm, $λ_{S} = {(λ_{p}^{- 1} - ν_{R})}^{- 1}$ = 1203 nm, and $λ_{S 2} = {(λ_{p}^{- 1} - 2 ν_{R})}^{- 1}$ = 1384 nm [8].

The external angles α and Δα controlled in the experiment are coupled by the law of refraction with the relative internal angles Θ_pm and Δβ_opt theoretically determined above. The angle α is defined between the pump input beam and the CaCO₃ crystal normal – see Fig. 3. The angle Δα is between the pump and probe input beams. These angles amount the values of $α = \arcsin (n_{p}^{o} \cdot \sin Θ_{pm})$ ≈11.5^о and $Δ α = α - \arcsin [n_{p}^{e} \cdot \sin (Θ_{pm} - Δ β_{opt})]$ ≈2.2^о for Θ_pm = 6.9^о, $n_{p}^{o}$ = 1.642, and $n_{p}^{e}$ = 1.641.

The laser was pumped by a laboratory-designed Nd:YAG oscillator-amplifier system generating horizontally-polarized pulses with a duration of τ_p = 6 ns (FWHM) at the wavelength of λ_p = 1064 nm in TEM00 mode (M² ≈1.1) with the maximum energy of W_p = 10 mJ at a repetition rate of 5 Hz. A birefringent prism was used to split the pump laser radiation into two orthogonally polarized beams (one represents the pump beam and the second the probe beam) and their energy ratio can be tuned by a half-wave plate (λ/2) placed in front of the birefringent prism. A wide-aperture objective with a focal length of 500 mm was used for focusing the parallel pump and probe beams into the active crystal of the parametric Raman laser. The pump laser beam radius at the focal plane was r₀ = 220 μm at the pump beam angular divergence of 3.4 mrad in air, which corresponds to its angular divergence of 2.1 mrad in the crystal medium, which is 3.3 times lower than tangential phase matching angular tolerance of ΔΘ_pm = 7 mrad (Fig. 2).

To control the incidence angle α of the pump beam into the active crystal, the active crystal was placed on a precise rotation stage with the rotation axis located in a plane of the active crystal input face. The crystal input face was also located at the pump and probe beams intersection point. The pump and probe beams should be parallel in front of the objective to get their intersection point precisely at the focal point behind the objective. A distance between the parallel pump and probe beams (in front of the objective) must be aligned precisely in order to get the adjusted incidence angle (α – Δα) of the probe beam and therefore the rotary mirror M was placed on a mirror mount with a precise transversal translation stage.

The Raman laser cavity was formed by a set of two similar concave mirrors with the radius of curvature of 500 mm aligned perpendicularly to the pump beam. The cavity mirrors were placed at a distance of 0.5 cm from the nearest active crystal face. The mirrors curvature radius of 500 mm was chosen experimentally as optimal from the available mirrors in order to achieve the mode matching between the pump beam and calculated Stokes resonator mode. We used two sets of the cavity mirrors. Both mirror sets had high transmission at the pump/probe (1064 nm) and anti-Stokes (954 nm) wavelengths (for its single-pass interaction) and high reflectivity at the 1^st Stokes (1203 nm) wavelength (for the lowest threshold of the extracavity SRS generation). We also studied competition between the anti-Stokes (954 nm) and the 2^nd Stokes (1384 nm) generation, and therefore the mirror sets 1 and 2 had respectively low and high transmissions at 1384 nm. Precisely, the mirror set 1 had transmissions of T = 72% @ 954 nm, T = 77% @ 1064 nm, T = 4% @ 1203 nm, and T = 0.5% @ 1384 nm, and the mirror set 2 had transmissions of T = 98% @ 954 nm, T = 98% @ 1064 nm, T = 2% @ 1203 nm, and T = 80% @ 1384 nm.

4. Mathematical modeling

In [8] we have developed the single-pass transient parametric Raman generation model differing from the conventional models by considering the orthogonally polarized four-wave mixing interaction. Here we consider a quasi-steady-state limit of this model suitable for nanosecond pumping, i.e. we neglect the temporal derivatives in the material equations substituting it in the wave equations and also we take the wave equations for the Stokes SRS components propagating not only in the forward direction (z), but also in the backward direction (– z) in the extracavity Raman laser:

\begin{array}{l} \frac{\partial E_{a S}}{\partial z} + \frac{n_{a S}^{e}}{c} \frac{\partial E_{a S}}{\partial t} = - \frac{g_{p}}{2} \frac{λ_{p}}{λ_{a S}} {| E_{p r o b e} |}^{2} E_{a S} - p E_{p r o b e} E_{p u m p} E_{S}^{(+) *} e^{i \cdot Δ k \cdot z}, \\ \frac{\partial E_{p r o b e}}{\partial z} + \frac{n_{p}^{e}}{c} \frac{\partial E_{p r o b e}}{\partial t} = \frac{g_{p}}{2} {| E_{a S} |}^{2} E_{p r o b e} + p E_{a S} E_{S}^{(+)} E_{p u m p}^{*} e^{- i \cdot Δ k \cdot z}, \\ \frac{\partial E_{p u m p}}{\partial z} + \frac{n_{p}^{o}}{c} \frac{\partial E_{p u m p}}{\partial t} = - \frac{g_{S}}{2} \frac{λ_{S}}{λ_{p}} ({| E_{S}^{(+)} |}^{2} + {| E_{S}^{(-)} |}^{2}) E_{p u m p} - p E_{S}^{(+)} E_{a S} E_{p r o b e}^{*} e^{- i \cdot Δ k \cdot z}, \\ \frac{\partial E_{S}^{(+)}}{\partial z} + \frac{n_{S}^{o}}{c} \frac{\partial E_{S}^{(+)}}{\partial t} = \frac{g_{S}}{2} {| E_{p u m p} |}^{2} E_{S}^{(+)} - \frac{g_{S 2}}{2} \frac{λ_{S 2}}{λ_{S}} ({| E_{S 2}^{(+)} |}^{2} + {| E_{S 2}^{(-)} |}^{2}) E_{S}^{(+)} + p E_{p u m p} E_{p r o b e} E_{a S}^{*} e^{i \cdot Δ k \cdot z}, \\ - \frac{\partial E_{S}^{(-)}}{\partial z} + \frac{n_{S}^{o}}{c} \frac{\partial E_{S}^{(-)}}{\partial t} = \frac{g_{S}}{2} {| E_{p u m p} |}^{2} E_{S}^{(-)} - \frac{g_{S 2}}{2} \frac{λ_{S 2}}{λ_{S}} ({| E_{S 2}^{(+)} |}^{2} + {| E_{S 2}^{(-)} |}^{2}) E_{S}^{(-)}, \\ \pm \frac{\partial E_{S 2}^{(\pm)}}{\partial z} + \frac{n_{S 2}^{o}}{c} \frac{\partial E_{S 2}^{(\pm)}}{\partial t} = \frac{g_{S 2}}{2} ({| E_{S}^{(+)} |}^{2} + {| E_{S}^{(-)} |}^{2}) E_{S 2}^{(\pm)}, \end{array}

where E_aS, E_probe, and E_pump are the slowly varying amplitudes of the anti-Stokes, probe, and pump waves propagating only in the forward direction;

E_{S}^{(\pm)}

and

E_{S 2}^{(\pm)}

are the slowly varying amplitudes of the 1^st and 2^nd Stokes SRS components propagating in the forward ( + ) and backward (–) directions in the Raman laser cavity; g_p, g_S, and g_S₂ are the Raman gain coefficients of the active crystal for the pump/probe, 1^st Stokes, and 2^nd Stokes waves respectively; p is the parametric four-wave-mixing coupling coefficient [14]; Δk is the wave mismatch of four-wave-mixing interaction of the anti-Stokes, probe, pump, and Stokes waves propagating in the forward direction. The boundary conditions can be written as

\begin{array}{l} E_{a S} (0, t) = 0, \\ E_{p r o b e} (0, t) = \sqrt{I_{p r o b e}^{i n} (t)}, \\ E_{p u m p} (0, t) = \sqrt{I_{p u m p}^{i n} (t)}, \\ E_{S, S 2}^{(+)} (0, t) = \sqrt{ε {| E_{p u m p} (0, t) |}^{2} + {| E_{S, S 2}^{(-)} (0, t - t_{i n}) |}^{2} T_{t r} R_{S, S 2}^{i n}}, \\ E_{S, S 2}^{(-)} (L, t) = \sqrt{ε {| E_{p u m p} (L, t - t_{t r}) |}^{2} + {| E_{S, S 2}^{(+)} (L, t - t_{o u t}) |}^{2} T_{t r} R_{S, S 2}^{o u t}}, \end{array}

where $R_{S, S 2}^{i n, o u t}$ is the reflection coefficient of the input (in) and output (out) cavity mirrors for the respective SRS components, $I_{p u m p}^{i n} (t)$ and $I_{p r o b e}^{i n} (t)$ are the temporal dependences of the pump and probe pulse intensity having a Gaussian shape with a duration of τ_p at the active crystal input, ε = 10 ^{– 13} is the seed coefficient [16], t_in and t_out are the times of double pass from the cavity mirrors to the nearest faces of the active crystal, t_tr is the transit time of the active crystal; T_tr is the single-pass transmission coefficient of the active crystal (the reflection from two faces and the internal losses of the active crystal are taken into account in this coefficient).

From the literature [17] we know the value of the CaCO₃ Raman gain coefficient of g ≈10 cm/GW at the excitation wavelength of λ_exc = 0.532·10 ^{– 4} cm for the SRS at the 1^st Stokes wavelength of $λ_{S R S} = {(λ_{e x c}^{- 1} - ν_{R})}^{- 1}$ = 0.565·10 ^{– 4} cm. This value was confirmed by us in [18] for this CaCO₃ sample used as the active crystal. The Raman gain should be decreasing with the λ_exc wavelength increase [19,20] and therefore it’s necessary to calculate the Raman gain coefficients g_p, g_S, and g_S₂ at the wavelengths of λ_p, λ_S, and λ_S₂, respectively, taking part in our interaction. We defined the Raman gain coefficient from the experimental measurement of the 1^st Stokes SRS generation threshold in the extracavity Raman laser according to the formula [21]:

g \approx \frac{25}{I_{e x c}^{t h} L} (\frac{L_{c a v}}{τ_{e x c} c} + \frac{1}{25} \ln \frac{1}{\sqrt{T_{t r}^{2} R_{S R S}^{i n} R_{S R S}^{o u t}}}),

where

I_{e x c}^{t h} = W_{e x c}^{t h} / τ_{e x c} π r_{e x c}^{2}

is the threshold excitation pulse intensity,

W_{e x c}^{t h}

is the threshold excitation pulse energy, τ_exc is the input excitation pulse duration, r_exc is the excitation beam radius in the active crystal, L is the active crystal length, L_cav is the optical length of the Raman laser external cavity,

R_{S R S}^{i n, o u t}

is the reflection coefficient of the input (in) and output (out) cavity mirrors at the SRS wavelength, c is the speed of light. In our laser setup (Fig. 3) at 1064-nm excitation with r_exc = 220 μm and

R_{S R S}^{i n, o u t}

= 98% we had the experimental SRS generation threshold at

W_{e x c}^{t h}

= 1.9 mJ and therefore the Raman gain for the 1203-nm 1^st Stokes SRS component was estimated to be g ≈2 cm/GW. Additionally, we experimentally defined the Raman gain coefficient of the CaCO₃ crystal under extracavity excitation by the Nd:YAG laser generating at the wavelength of λ_exc = 1.319·10 ^{– 4} cm. According to the formula (3) we calculated the value of g ≈0.3 cm/GW for

λ_{S R S} = {(λ_{e x c}^{- 1} - ν_{R})}^{- 1}

= 1.540·10 ^{– 4} cm. Thus, we obtained three measurement points of g ≈10, 2, and 0.3 cm/GW for λ_SRS = 0.565·10 ^{– 4}, 1.203·10 ^{– 4}, and 1.540·10 ^{– 4} cm, respectively. Therefore, the Raman gain dependence on the wavenumber (

λ_{S R S}^{- 1}

) in the wavelength range from λ_SRS = 0.565·10 ^{– 4} cm to λ_SRS = 1.540·10 ^{– 4} cm was found to be close to linear amounting

g \approx 8.6 \cdot 10^{- 13} \cdot λ_{S R S}^{- 1} - 5.2 \cdot 10^{- 9}

(in cm/W). Hence, we have calculated the values of g_p ≈2.8 cm/GW, g_S ≈2 cm/GW and g_S₂ ≈1 cm/GW at λ_SRS = λ_p, λ_S, and λ_S₂, respectively, that were used for the modeling.

The parametric coupling coefficient (p) was variable at the modeling. The best agreement of the modeling results (lines) with the experimental results (points) shown in Figs. 4 and 5 took place at p = g_p/2. It confirms the theoretical conclusion that the parametric coupling coefficient should be close to the Raman coupling coefficient [8,14] considering the fact that the Raman gain coefficient decreases with the wavelength [19,20]. Therefore, the parametric coefficient should be defined by the Raman gain (g_p) for the wavelength λ_p intermediate between the parametrically coupled anti-Stokes (λ_aS) and Stokes (λ_S) wavelengths. So, the following results are presented at the fixed value of p = g_p/2. Furthermore, at the modeling we used the fixed values of λ_p = 1064 nm, τ_p = 6 ns, r_p = 220 μm, L = 3.2 cm, T_tr = 80%, $R_{S}^{i n}$ = $R_{S}^{o u t}$ = 98%, and Δk = 0.

Fig. 4 The calculated (lines) and experimental (points) dependences of conversion efficiency η_aS into the anti-Stokes wave from the overall (probe + pump) input radiation on the ratio W_probe/W_pump between the probe and pump input pulse energies at various overall input pulse energies of W_probe + W_pump = 8.1, 5, and 3.2 mJ for (a) $R_{S 2}^{i n}$ = $R_{S 2}^{o u t}$ = 99% and (b) $R_{S 2}^{i n}$ = $R_{S 2}^{o u t}$ = 20%.

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Fig. 5 The calculated (lines) and experimental (points) optimization of the cavity mirror reflection at the 2^nd Stokes wavelength $R_{S 2}^{i n}$ = $R_{S 2}^{o u t}$ for various overall input pulse energies of W_probe + W_pump = 8.1, 5, and 3.2 mJ at the optimal probe/pump ratio (W_probe/W_pump)_opt = 0.3.

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Fig. 4 shows the computational optimization (lines) of a ratio W_probe/W_pump between the probe and pump input pulse energies for two values of $R_{S 2}^{i n}$ = $R_{S 2}^{o u t}$ = 99% (Fig. 4(a)) and $R_{S 2}^{i n}$ = $R_{S 2}^{o u t}$ = 20% (Fig. 4(b)) at various overall input pulse energies of W_probe + W_pump = 8.1, 5, and 3.2 mJ.

This figure demonstrates not only the optimal ratio (W_probe /W_pump)_opt = 0.3 independent on the overall input pulse energy, but also the importance of decreasing the cavity mirror reflection at the 2^nd Stokes wavelength (from 99% down to 20%) in order to increase the conversion efficiency η_aS into the anti-Stokes wave from the overall (probe + pump) input radiation. At decreasing the $R_{S 2}^{i n, o u t}$ the anti-Stokes output was maximized at the optimum value of $R_{S 2}^{i n, o u t}$ = 20 - 30% shown in Fig. 5. Existence of the optimum $R_{S 2}^{i n, o u t}$ needs to be explained because at the first sight the best case should be $R_{S 2}^{i n, o u t}$ = 0 to suppress the 2^nd Stokes generation for increasing the efficiency of the anti-Stokes generation, but the competition between the generated waves is more difficult.

Fig. 6 demonstrates the results of the numerical simulation of the wave interaction in the extracavity parametric Raman CaCO₃ laser at the optimal probe/pump ratio (0.3) for the overall input pulse energy of 8.1 mJ and various values of $R_{S 2}^{i n, o u t}$ . Here the 1^st and 2^nd Stokes output intensities generating in the Raman laser external cavity are the sum of all the cavity outputs from both the cavity mirrors (at its transmission) and from the faces of the active crystal (at its Fresnel reflection).

Fig. 6 Results of the numerical simulation of the wave interaction in the extracavity parametric Raman CaCO₃ laser at W_probe + W_pump = 8.1 mJ and (W_probe/W_pump)_opt = 0.3.

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It is necessary to note that we used here the high-Q external cavity for the Stokes wave to decrease the SRS generation threshold in the short 3.2-cm CaCO₃ Raman crystal lower than the crystal optical damage threshold that was measured in our experiment to be about 1 GW/cm² at the pulse duration of 6 ns. However, using the high-Q cavity presents a challenge for efficient anti-Stokes four-wave-mixing generation because the pump intensity taking part in this four-wave mixing became close to zero due to full depletion at SRS conversion into the Stokes radiation. Therefore we had low conversion efficiency into the anti-Stokes wave of η_aS = 2.6% in the case of $R_{S 2}^{i n, o u t}$ = 1% (Fig. 6(a)) where the 1^st Stokes generation was the most efficient due to full depletion of pumping and prevention of the efficient 2^nd Stokes generation. On contrary, at high reflection $R_{S 2}^{i n, o u t}$ = 99% (Fig. 6(c)) the anti-Stokes conversion stopped due to full depletion of the 1^st Stokes intensity at its conversion into the 2^nd Stokes component led to even lower anti-Stokes generation efficiency of η_aS = 1.2%. The optimal case of $R_{S 2}^{i n, o u t}$ = 20% shown in Fig. 6(b) corresponds to the situation where all the waves taking part in the four-wave-mixing anti-Stokes generation had no full depletion that gave us the maximal value of the anti-Stokes generation efficiency of η_aS = 3.9% at the given level of excitation (8.1 mJ).

5. Experimental study

Using the rotation stage for the CaCO₃ crystal and translation stage for the mirror M (Fig. 3) allowed to adjust the optimal angles of α_opt = 11.5^о and Δα_opt = 2.2^о for the phase matching fulfilment and walk-off compensation, respectively. Fig. 7 shows the angular mismatch characteristics for the output anti-Stokes pulse energy W_aS in dependence on the angles α and Δα around α_opt and Δα_opt at W_probe + W_pump = 5 mJ and W_probe /W_pump = 0.3 for the mirror set 2.

Fig. 7 The angular mismatch characteristics for the anti-Stokes pulse energy W_aS in dependence on the angles α (a) and Δα (b) (initial values α_opt = 11.5^о, Δα_opt = 2.2^о) at W_probe + W_pump = 5 mJ and W_probe /W_pump = 0.3 for the mirror set 2.

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In Fig. 7(a) the phase matching angle (α) mismatch linewidth (FWHM) is 0.65°. This value is in an agreement with the theoretical value of ΔΘ_{pm $n_{a S}^{e}$} = 0.658° ( $n_{a S}^{e}$ = 1.64) where ΔΘ_pm = 7 mrad is the tangential phase matching angular tolerance inside the crystal medium (Fig. 2(a)). In Fig. 7(b) the walk-off compensation angle (Δα) mismatch linewidth (FWHM) is 0.8° in an agreement with the theoretical line ΔΘ_pm(Δβ) width around the tangential phase matching (Fig. 2(a)).

In these optimal conditions all the interacting radiation components propagated coaxially in the CaCO₃ crystal medium, but at the laser output there were two separated radiation beams (channels) with the Δα_opt angle between them (see also Fig. 3). A pump output channel containing not only the pump, but also the 1^st and 2^nd Stokes components; a probe output channel containing not only the probe, but also the anti-Stokes component. Fig. 8 demonstrates the radiation spectra in the probe and pump channels measured at two separate outputs of the laser (Fig. 3).

Fig. 8 Output radiation spectra in the probe and pump channels described in Fig. 3.

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The results of experimental optimization (rotating the λ/2 plate shown in Fig. 3) of the probe/pump ratio W_probe /W_pump were presented in Fig. 4 in comparison with the modeling results. Note that we had an agreement of the modeling (lines) with the experiment (points) not only in the maximum values of η_aS at the optimum ratio (W_probe /W_pump)_opt = 0.3, but also in the linewidth of the optimum ratio mismatch. Some disagreement in the maximum values of η_aS was only at $R_{S 2}^{i n}$ = $R_{S 2}^{o u t}$ = 99% (Fig. 4(a)) that can be explained by higher efficiency of the 2^nd Stokes generation in the experiment than in the modeling due to weak parametric coupling of the 2^nd Stokes wave with the pump that was unaccounted at the modeling. At decreasing the $R_{S 2}^{i n, o u t}$ value this disagreement disappeared, and the anti-Stokes generation efficiency was maximized at $R_{S 2}^{i n, o u t}$ = 20% amounting η_aS = 3.7% (in the experiment) at the highest overall (probe + pump) input energy of 8.1 mJ (Figs. 4 and 5). Note also that the experimental result of the cavity mirrors optimization, where the mirror set 2 is non-optimal, will remain unchanged if we use equally high transmission at 954 and 1064 nm in both the mirror sets because it can improve the result for the mirror set 2 not higher than 2 times, but the results obtained for the mirror sets 1 and 2 differ from each other much more.

Figure 9 shows the oscillograms of the input probe and output anti-Stokes pulses for the mirror set 1 (Fig. 9(a)) and mirror set 2 (Fig. 9(b)) at W_probe + W_pump = 5 mJ and W_probe /W_pump = 0.3. The output pulse has temporal modulation with the period equal to the cavity round-trip time. This modulation characterizes the SRS generation of the Stokes radiation in the high-Q cavity transferred into the anti-Stokes pulse due to Stokes-anti-Stokes parametric coupling. In Fig. 9(a) the anti-Stokes pulse has a short duration of about 1 ns with only one intensive peak of this modulation, but in Fig. 9(b) it is substantially longer of about 3 ns with several peaks of these modulations. This is in agreement with the modeling results (Fig. 6) where the anti-Stokes pulse duration is shortened at $R_{S 2}^{i n, o u t}$ = 99% (Fig. 6(c)) because of the full depletion of the 1^st Stokes component at its SRS conversion to the 2^nd Stokes component, but the anti-Stokes pulse duration is the longest and its generation efficiency is the highest at the optimal value of $R_{S 2}^{i n, o u t}$ = 20% (Fig. 6(b)) due to prevention of full depletion of the pump and 1^st Stokes pulses (Fig. 6(b)).

Fig. 9 The oscillograms of the input probe and output anti-Stokes pulses at W_probe + W_pump = 5 mJ and optimal probe/pump ratio W_probe /W_pump = 0.3 for the mirror set 1 (a) and 2 (b).

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Fig. 10 shows the output anti-Stokes pulse energy versus the overall (probe + pump) input pulse energy with both cavity mirror sets at the optimal probe/pump ratio. The output energy growth for the anti-Stokes radiation was close to linear. It can be explained by the fact that due to parametric Stokes ↔ anti-Stokes coupling the anti-Stokes component follows the 1^st Stokes component having similarly linear energy growth at its SRS generation in the external cavity. In agreement with the modeling results, the highest output 954-nm anti-Stokes pulse energy was achieved using the mirror set 2. The highest output energy was 0.3 mJ at the input energy of 8.1 mJ corresponding to the highest optical-to-optical efficiency of 3.7% and slope efficiency of 5.4%. The experimental value of the anti-Stokes generation efficiency of 3.7% can be increased up to 4% taking into account the partial transmission of the active crystal output face and the cavity output mirror at 954 nm of 95% and 98%, respectively. So, we have achieved a good agreement with the modeling result (3.9%) obtained due to careful mode matching by selection of the cavity mirrors curvature radii from the available collection of 100 mm, 250 mm, 500 mm, 1000 mm, and flat (optimal value was 500 mm). It allowed not only the output energy characteristics improvement, but also the output beam quality. Inset in Fig. 10 demonstrates the 954-nm anti-Stokes beam profile close to a fundamental Gaussian mode. The measured beam quality M² was 1.1 × 1.2 in the horizontal and vertical axis, respectively.

Fig. 10 The Anti-Stokes pulse energy as a function of the overall (probe + pump) input pulse energy at the optimal probe/pump ratio. Inset shows the anti-Stokes beam profile and the measured beam quality close to ideal.

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6. Conclusions

In conclusion, the extracavity parametric Raman CaCO₃ anti-Stokes laser at 954 nm excited by two orthogonally polarized (pump and probe) equal frequency laser beams was proposed and investigated. Tangential phase matching conditions for the orthogonally polarized Raman components four-wave mixing in the CaCO₃ crystal at SRS under 1064-nm excitation were theoretically studied and applied for the novel parametric Raman laser allowing to increase the 954-nm anti-Stokes generation efficiency. In agreement with the theoretical modeling, in our experimental results we have achieved the highest optical-to-optical (1064 nm to 954 nm) efficiency of 4%. The anti-Stokes output pulse energy was 0.3 mJ with high beam quality factor M² = 1.1 × 1.2. To our best knowledge, this is the highest anti-Stokes generation efficiency ever achieved in the crystalline parametric Raman anti-Stokes lasers. This result is not only due to the increased excitation wavelength (1064 nm) than in our previous work (532 nm) [8] giving wider angular tolerance of tangential phase marching (7 mrad against 4.7 mrad [8]), but also because of control by competition between generation of Raman components of several orders in the Raman laser cavity at quasi-steady-state SRS regime. We studied the case of the high-Q external cavity for the 1^st Stokes component which was good for the lowest SRS generation threshold, but it led to limitation of anti-Stokes generation efficiency due to full depletion of pumping. Therefore, the cavity mirror optimization for the 2^nd Stokes component was required in order to get the interacting wave competition advantageous for the anti-Stokes generation.

As for the application aspect, frequency doubling of the 954-nm output radiation (it was also successfully tested within a frame of this work using a KDP crystal) allows to generate nanosecond blue radiation at a wavelength of 477 nm spectrally located at the minimum absorption of a water that might be interesting for subsea optical communications.

Funding

Czech Science Foundation - Project No. 18-11954S.

References and links

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Extracavity pumped parametric Raman nanosecond crystalline anti-Stokes laser at 954 nm with collinear orthogonally polarized beam interaction at tangential phase matching

Abstract

1. Introduction

2. Tangential phase matching fulfillment

3. Laser system schematic

4. Mathematical modeling

5. Experimental study

6. Conclusions

Funding

References and links

Cited By

Figures (10)

Equations (3)

Optics Express